Hostname: page-component-84b7d79bbc-g78kv Total loading time: 0 Render date: 2024-08-01T08:38:00.640Z Has data issue: false hasContentIssue false
  • English
  • Français

Dissipative operators and series inequalities

Published online by Cambridge University Press:  17 April 2009

Herbert A. Gindler  and
Jerome A. Goldstein
Herbert A. Gindler
Affiliation:
Department of Mathematics, San Diego State University, San Diego, California 92182, USA
Jerome A. Goldstein
Affiliation:
Department of Mathematics, Tulane University, New Orleans, Louisiana 70118, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Of concern is the best constant K in the inequality ‖Ax‖2 ≤ K‖A2x‖‖x‖ where A generates a strongly continuous contraction semigroup in a Hilbert space. Criteria are obtained for approximate extremal vectors x when K = 2 (K ≤ 2 always holds). By specializing A + I to be a shift operator on a sequence space, very simple proofs of Copson's recent results on series inequalities follow. Inequalities of the above type are also studied on LP spaces, and earlier results of the authors and of Holbrook are improved.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 23 , Issue 3 , June 1981 , pp. 429 - 442
Copyright
Copyright © Australian Mathematical Society 1981

References

[1]Clarkson, James A., “Uniformly convex spaces”, Trans. Amer. Math. Soc. 40 (1936), 396–414.CrossRefGoogle Scholar
[2]Copson, E.T., “Two series inequalities”, Proc. Roy. Soc. Edinburgh Sect. A 83 (1979), 109–114.CrossRefGoogle Scholar
[3]Gindler, Herbert A. and Goldstein, Jerome A., “Dissipative operator versions of some classical inequalities”, J. Analyse Math. 28 (1975), 213–238.CrossRefGoogle Scholar
[4]Holbrook, John A.R., “A Kallman-Rota inequality for nearly Euclidean spaces”, Adv. in Math. 14 (1974), 335–345.CrossRefGoogle Scholar
[5]Kallman, Robert A. and Rota, Gian-Carlo, “On the inequality ||ƒ′||2 ≥ 4||ƒ|| · ||ƒ″||, Inequalities, II, 187–192 (Proc. Second Sympos., US Air Force Acad., Colorado, 1967. Academic Press, New York, 1970).Google Scholar
[6]Kato, Tosio, “On an inequality of Hardy, Littlewood, and Pòlya”, Adv. in Math. 7 (1971), 217–218.CrossRefGoogle Scholar
[7]Kwong, Man Kam and Zetti, A., “Landau's inequality”, submitted.Google Scholar
[8]Kwong, Man Kam and Zetti, A., “Ramifications of Landau's inequality”, Proc. Roy. Soc. Edinburgh Sect. A 86 (1980), 175–212.CrossRefGoogle Scholar
[9] Ю.И. Лобич [Ljubič, Ju. I.] “О неравенствах между степеннми линейного опе-раеора” [On inequalities between the powers of a linear operator], Izv. Akad. Nauk SSSF Ser. Mat. 24 (1960), 825–864; English transl: Transl. Amer. Math. Soc. (2) 40 (1964), 39–84.Google Scholar
[10]Lumer, G. and Phillips, R.S., “Dissipative operators in a Banach space”, Pacific J. Math. 11 (1961), 679–698.CrossRefGoogle Scholar